A question on the ultrafilter numberSurjective Maps onto $aleph$-numbersMaximal chains in a quasi-order of linear order typesIs there a modification of Martin's Axiom which admits non-measurable sets of size less than continuum?What sort of large cardinal can continuum be?Is there a locally compact, $omega_1$-compact, not $sigma$-countably compact space of size $aleph_1$?Is there nontrivial structure to forcing axioms?The minimum cardinality of an almost disjoint reaping familyThe “strong” measure numberTranslates of measure zero setThe Parovichenko cardinal, is it equal to $maxaleph_2,mathfrak p$?

A question on the ultrafilter number


Surjective Maps onto $aleph$-numbersMaximal chains in a quasi-order of linear order typesIs there a modification of Martin's Axiom which admits non-measurable sets of size less than continuum?What sort of large cardinal can continuum be?Is there a locally compact, $omega_1$-compact, not $sigma$-countably compact space of size $aleph_1$?Is there nontrivial structure to forcing axioms?The minimum cardinality of an almost disjoint reaping familyThe “strong” measure numberTranslates of measure zero setThe Parovichenko cardinal, is it equal to $maxaleph_2,mathfrak p$?













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Let $mathfraku$ denote the ultrafilter number, which is defined to be the minimum cardinality of a subset of $mathcalP(mathbb N)$ which is a base for a nonprincipal ultrafilter on $mathbbN$. Clearly $aleph_1leq frakuleq 2^aleph_0$, so it is only interesting to study $fraku$ under the negation of CH. Kunen proved that it is consistent that CH fails and that $fraku=aleph_1$. Martin's axiom implies that $fraku=2^aleph_0$.



Is it consistent that $aleph_1<fraku<2^aleph_0$? If so, can I please have a reference?










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$endgroup$
















    5












    $begingroup$


    Let $mathfraku$ denote the ultrafilter number, which is defined to be the minimum cardinality of a subset of $mathcalP(mathbb N)$ which is a base for a nonprincipal ultrafilter on $mathbbN$. Clearly $aleph_1leq frakuleq 2^aleph_0$, so it is only interesting to study $fraku$ under the negation of CH. Kunen proved that it is consistent that CH fails and that $fraku=aleph_1$. Martin's axiom implies that $fraku=2^aleph_0$.



    Is it consistent that $aleph_1<fraku<2^aleph_0$? If so, can I please have a reference?










    share|cite|improve this question









    $endgroup$














      5












      5








      5





      $begingroup$


      Let $mathfraku$ denote the ultrafilter number, which is defined to be the minimum cardinality of a subset of $mathcalP(mathbb N)$ which is a base for a nonprincipal ultrafilter on $mathbbN$. Clearly $aleph_1leq frakuleq 2^aleph_0$, so it is only interesting to study $fraku$ under the negation of CH. Kunen proved that it is consistent that CH fails and that $fraku=aleph_1$. Martin's axiom implies that $fraku=2^aleph_0$.



      Is it consistent that $aleph_1<fraku<2^aleph_0$? If so, can I please have a reference?










      share|cite|improve this question









      $endgroup$




      Let $mathfraku$ denote the ultrafilter number, which is defined to be the minimum cardinality of a subset of $mathcalP(mathbb N)$ which is a base for a nonprincipal ultrafilter on $mathbbN$. Clearly $aleph_1leq frakuleq 2^aleph_0$, so it is only interesting to study $fraku$ under the negation of CH. Kunen proved that it is consistent that CH fails and that $fraku=aleph_1$. Martin's axiom implies that $fraku=2^aleph_0$.



      Is it consistent that $aleph_1<fraku<2^aleph_0$? If so, can I please have a reference?







      set-theory lo.logic






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      asked 9 hours ago









      IsaacIsaac

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          10












          $begingroup$

          The answer to your question is yes. In fact, one can force to make $mathfrak u$ equal to any prescribed uncountable regular cardinal while making the cardinal of the continuum equal to any larger prescribed uncountable regular cardinal. This is proved in and old paper by Shelah and me:



          Blass, Andreas(1-PAS); Shelah, Saharon(1-RTG)
          Ultrafilters with small generating sets.
          Israel J. Math. 65 (1989), no. 3, 259–271.






          share|cite|improve this answer









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          • $begingroup$
            Thanks Andreas.
            $endgroup$
            – Isaac
            6 hours ago










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          10












          $begingroup$

          The answer to your question is yes. In fact, one can force to make $mathfrak u$ equal to any prescribed uncountable regular cardinal while making the cardinal of the continuum equal to any larger prescribed uncountable regular cardinal. This is proved in and old paper by Shelah and me:



          Blass, Andreas(1-PAS); Shelah, Saharon(1-RTG)
          Ultrafilters with small generating sets.
          Israel J. Math. 65 (1989), no. 3, 259–271.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Thanks Andreas.
            $endgroup$
            – Isaac
            6 hours ago















          10












          $begingroup$

          The answer to your question is yes. In fact, one can force to make $mathfrak u$ equal to any prescribed uncountable regular cardinal while making the cardinal of the continuum equal to any larger prescribed uncountable regular cardinal. This is proved in and old paper by Shelah and me:



          Blass, Andreas(1-PAS); Shelah, Saharon(1-RTG)
          Ultrafilters with small generating sets.
          Israel J. Math. 65 (1989), no. 3, 259–271.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Thanks Andreas.
            $endgroup$
            – Isaac
            6 hours ago













          10












          10








          10





          $begingroup$

          The answer to your question is yes. In fact, one can force to make $mathfrak u$ equal to any prescribed uncountable regular cardinal while making the cardinal of the continuum equal to any larger prescribed uncountable regular cardinal. This is proved in and old paper by Shelah and me:



          Blass, Andreas(1-PAS); Shelah, Saharon(1-RTG)
          Ultrafilters with small generating sets.
          Israel J. Math. 65 (1989), no. 3, 259–271.






          share|cite|improve this answer









          $endgroup$



          The answer to your question is yes. In fact, one can force to make $mathfrak u$ equal to any prescribed uncountable regular cardinal while making the cardinal of the continuum equal to any larger prescribed uncountable regular cardinal. This is proved in and old paper by Shelah and me:



          Blass, Andreas(1-PAS); Shelah, Saharon(1-RTG)
          Ultrafilters with small generating sets.
          Israel J. Math. 65 (1989), no. 3, 259–271.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 7 hours ago









          Andreas BlassAndreas Blass

          58k7138225




          58k7138225











          • $begingroup$
            Thanks Andreas.
            $endgroup$
            – Isaac
            6 hours ago
















          • $begingroup$
            Thanks Andreas.
            $endgroup$
            – Isaac
            6 hours ago















          $begingroup$
          Thanks Andreas.
          $endgroup$
          – Isaac
          6 hours ago




          $begingroup$
          Thanks Andreas.
          $endgroup$
          – Isaac
          6 hours ago

















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